Skip to main content
Log in

Emergence of Oscillations in a Mixed-Mechanism Phosphorylation System

  • Published:
Bulletin of Mathematical Biology Aims and scope Submit manuscript

Abstract

This work investigates the emergence of oscillations in one of the simplest cellular signaling networks exhibiting oscillations, namely the dual-site phosphorylation and dephosphorylation network (futile cycle), in which the mechanism for phosphorylation is processive, while the one for dephosphorylation is distributive (or vice versa). The fact that this network yields oscillations was shown recently by Suwanmajo and Krishnan. Our results, which significantly extend their analyses, are as follows. First, in the three-dimensional space of total amounts, the border between systems with a stable versus unstable steady state is a surface defined by the vanishing of a single Hurwitz determinant. Second, this surface consists generically of simple Hopf bifurcations. Next, simulations suggest that when the steady state is unstable, oscillations are the norm. Finally, the emergence of oscillations via a Hopf bifurcation is enabled by the catalytic and association constants of the distributive part of the mechanism; if these rate constants satisfy two inequalities, then the system generically admits a Hopf bifurcation. Our proofs are enabled by the Routh–Hurwitz criterion, a Hopf bifurcation criterion due to Yang, and a monomial parametrization of steady states.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

Notes

  1. Network (1) is symmetric to the mixed-mechanism network in which phosphorylation is distributive (instead of processive) and dephosphorylation is processive (instead of distributive), so our results apply equally well to that network (cf. Suwanmajo and Krishnan 2015, networks 21, 22).

  2. This file and others mentioned below are in the Supporting Information; see “Appendix A”.

  3. The functions are provided as a text file in the Supporting Information. See “Appendix A”.

References

  • Aoki K, Yamada M, Kunida K, Yasuda S, Matsuda M (2011) Processive phosphorylation of ERK MAP kinase in mammalian cells. Proc Natl Acad Sci USA 108(31):12675–12680

    Article  Google Scholar 

  • Atkins P, De Paula J, Keeler J (2018) Atkins’ physical chemistry. Oxford University Press, Oxford

    Google Scholar 

  • Bure EG, Rozenvasser YN (1974) On investigations of autooscillating system sensitivity, no 7. Avtomat. i Telemekh, pp 9–17

  • Conradi C, Mincheva M (2014) Catalytic constants enable the emergence of bistability in dual phosphorylation. J R Soc Interface 11(95):20140158

    Article  Google Scholar 

  • Conradi C, Shiu A (2015) A global convergence result for processive multisite phosphorylation systems. Bull Math Biol 77(1):126–155

    Article  MathSciNet  MATH  Google Scholar 

  • Conradi C, Shiu A (2018) Dynamics of post-translational modification systems: recent progress and future challenges. Biophys J 114(3):507–515

    Article  Google Scholar 

  • Conradi C, Feliu E, Mincheva M, Wiuf C (2017) Identifying parameter regions for multistationarity. PLoS Comput Biol 13(10):e1005751

    Article  Google Scholar 

  • Dhooge A, Govaerts W, Kuznetsov YA (2003) MATCONT: a MATLAB package for numerical bifurcation analysis of ODEs. ACM Trans Math Softw 29(2):141–164

    Article  MathSciNet  MATH  Google Scholar 

  • Domijan M, Kirkilionis M (2009) Bistability and oscillations in chemical reaction networks. J Math Biol 59(4):467–501

    Article  MathSciNet  MATH  Google Scholar 

  • Eithun M, Shiu A (2017) An all-encompassing global convergence result for processive multisite phosphorylation systems. Math Biosci 291:1–9

    Article  MathSciNet  MATH  Google Scholar 

  • Errami H, Eiswirth M, Grigoriev D, Seiler WM, Sturm T, Weber A (2015) Detection of Hopf bifurcations in chemical reaction networks using convex coordinates. J Comput Phys 291:279–302

    Article  MathSciNet  MATH  Google Scholar 

  • Ferrell JE, Ha SH (2014) Ultrasensitivity part II: multisite phosphorylation, stoichiometric inhibitors, and positive feedback. Trends Biochem Sci 39(11):556–569

    Article  Google Scholar 

  • Gantmacher FR (1959) Matrix theory, vol 21. Chelsea, New York

    MATH  Google Scholar 

  • Gatermann K, Eiswirth M, Sensse A (2005) Toric ideals and graph theory to analyze Hopf bifurcations in mass action systems. J Symb Comput 40(6):1361–1382

    Article  MathSciNet  MATH  Google Scholar 

  • Gelfand IM, Kapranov MM, Zelevinsky AV (1994) Discriminants, resultants and multidimensional determinants. Birkhäuser, Basel

    Book  MATH  Google Scholar 

  • Guckenheimer J, Holmes P (2013) Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, vol 42. Springer, Berlin

    MATH  Google Scholar 

  • Hadač O, Muzika F, Nevoral V, Přibyl M, Schreiber I (2017) Minimal oscillating subnetwork in the Huang-Ferrell model of the MAPK cascade. PLoS ONE 12(6):1–25

    Google Scholar 

  • Hell J, Rendall AD (2015) A proof of bistability for the dual futile cycle. Nonlinear Anal Real 24:175–189

    Article  MathSciNet  MATH  Google Scholar 

  • Hilioti Z, Sabbagh W, Paliwal S, Bergmann A, Goncalves MD, Bardwell L, Levchenko A (2008) Oscillatory phosphorylation of yeast Fus3 MAP kinase controls periodic gene expression and morphogenesis. Curr Biol 18(21):1700–1706

    Article  Google Scholar 

  • Hu H, Goltsov A, Bown JL, Sims AH, Langdon SP, Harrison DJ, Faratian D (2013) Feedforward and feedback regulation of the MAPK and PI3K oscillatory circuit in breast cancer. Cell Signal 25(1):26–32

    Article  Google Scholar 

  • Ingalls BP (2004) Autonomously oscillating biochemical systems: parametric sensitivity of extrema and period. Syst Biol 1(1):62–70

    Article  MathSciNet  Google Scholar 

  • Ingalls B, Mincheva M, Roussel MR (2017) Parametric sensitivity analysis of oscillatory delay systems with an application to gene regulation. Bull Math Biol 79(7):1539–1563

    Article  MathSciNet  MATH  Google Scholar 

  • Johnston MD (2014) Translated chemical reaction networks. Bull Math Biol 76(6):1081–1116

    Article  MathSciNet  MATH  Google Scholar 

  • Johnston MD, Müller S, Pantea C (2018) A deficiency-based approach to parametrizing positive equilibria of biochemical reaction systems. Preprint. arXiv:1805.09295

  • Liu WM (1994) Criterion of Hopf bifurcations without using eigenvalues. J Math Anal Appl 182(1):250–256

    Article  MathSciNet  MATH  Google Scholar 

  • Lozada-Cruz G (2012) The simple application of the implicit function theorem, vol XIX, no 1. Boletin de la Asociatión Matemática Venezolana

  • Millán MP, Dickenstein A (2018) The structure of MESSI biological systems. SIAM J Appl Dyn Syst 17(2):1650–1682

    Article  MathSciNet  MATH  Google Scholar 

  • Millán MP, Turjanski AG (2015) MAPK’s networks and their capacity for multistationarity due to toric steady states. Math Biosci 262:125–137

    Article  MathSciNet  MATH  Google Scholar 

  • Millán MP, Dickenstein A, Shiu A, Conradi C (2012) Chemical reaction systems with toric steady states. Bull Math Biol 74(5):1027–1065

    Article  MathSciNet  MATH  Google Scholar 

  • Müller S, Feliu E, Regensburger G, Conradi C, Shiu A, Dickenstein A (2016) Sign conditions for injectivity of generalized polynomial maps with applications to chemical reaction networks and real algebraic geometry. Found Comput Math 16(1):69–97

    Article  MathSciNet  MATH  Google Scholar 

  • Ode KL, Ueda HR (2017) Design principles of phosphorylation-dependent timekeeping in eukaryotic circadian clocks. Cold Spring Harb Perspect Biol 10:a028357

    Article  Google Scholar 

  • Rao S (2017) Global stability of a class of futile cycles. J Math Biol 74:709–726

    Article  MathSciNet  MATH  Google Scholar 

  • Rao S (2018) Stability analysis of the Michaelis-Menten approximation of a mixed mechanism of a phosphorylation system. Math Biosci 301:159–166

    Article  MathSciNet  MATH  Google Scholar 

  • Rubinstein BY, Mattingly HH, Berezhkovskii AM, Shvartsman SY (2016) Long-term dynamics of multisite phosphorylation. Mol Biol Cell 27(14):2331–2340

    Article  Google Scholar 

  • Salazar C, Höfer T (2009) Multisite protein phosphorylation—from molecular mechanisms to kinetic models. FEBS J 276(12):3177–3198

    Article  Google Scholar 

  • Suwanmajo T, Krishnan J (2015) Mixed mechanisms of multi-site phosphorylation. J R Soc Interface 12(107):20141405

    Article  Google Scholar 

  • Thomson M, Gunawardena J (2009) The rational parameterisation theorem for multisite post-translational modification systems. J Theor Biol 261(4):626–636

    Article  MathSciNet  MATH  Google Scholar 

  • Tung H-R (2018) Precluding oscillations in Michaelis-Menten approximations of dual-site phosphorylation systems. Math Biosci 306:56–59

    Article  MathSciNet  MATH  Google Scholar 

  • Virshup DM, Forger DB (2009) Keeping the beat in the rising heat. Cell 137(4):602–604

  • Wolfram Research Inc. (2018) Mathematica, Version 11.3, Champaign, IL

  • Yang X (2002) Generalized form of Hurwitz-Routh criterion and Hopf bifurcation of higher order. Appl Math Lett 15(5):615–621

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

AS was partially supported by the NSF (DMS-1312473/1513364 and DMS-1752672) and the Simons Foundation (#521874). AS thanks Jonathan Tyler for helpful discussions. CC was partially supported by the Deutsche Forschungsgemeinschaft DFG (DFG-284057449).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Maya Mincheva.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Electronic supplementary material

A Files in the Supporting Information

A Files in the Supporting Information

The following files can be found as supplementary material:

Text files:

  • mixed_H5N_kb.txt ...contains H5N, the numerator of \(\det H_5\) under the assumption \(k_2=k_6=k_9=k_b\)

  • mixed_W.txt ...contains a matrix W that defines (3)

  • mixed_xt.txt ...contains xt, the parameterization (7)

  • mixed_Jx.txt ...contains Jx, the Jacobian evaluated at the parameterization (7)

Mathematica Notebooks:

  • mixed_analysis_H5N_x1_LT.nb:

    Functionality: This file can be used to obtain \(\text {numerator}(\det H_5)\) as in (12), in particular to examine the coefficients \(\alpha _{01}\), \(\alpha _{10}\), ...

    Input: the file mixed_H5N_kb.txt

  • mixed_analysis_H5N_x2_LT.nb:

    Functionality: This file can be used to obtain \(\text {numerator}(\det H_5)\) as in (14), in particular to examine the coefficients \(\alpha _{0}\), ..., \(\alpha _{3}\) and \(\beta _0\), ..., \(\beta _3\).

    Input: the file mixed_H5N_kb.txt

  • mixed_coeffs_charpoly.nb:

    Functionality: This file can be used to obtain the characteristic polynomial of the Jacobian of the system (2). It contains the Mathematica commands to establish \(b_i>0\).

    Input: the file mixed_Jx.txt

  • mixed_Hi.nb:

    Functionality: This file can be used to obtain the determinants of the Hurwitz matrices \(H_2\), ..., \(H_5\). It contains the Mathematica commands to establish \(\det H_i >0\), for \(i=2\), 3, 4 and that \(\det H_5\) is of mixed sign.

    Input: the file mixed_Jx.txt

  • mixed_generate_rc.nb:

    Functionality: This file contains a realization of Procedure 5.1.

    Input: the files mixed_H5N_kb.txt, mixed_W.txt, mixed_xt.txt, mixed_Jx.txt.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Conradi, C., Mincheva, M. & Shiu, A. Emergence of Oscillations in a Mixed-Mechanism Phosphorylation System. Bull Math Biol 81, 1829–1852 (2019). https://doi.org/10.1007/s11538-019-00580-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11538-019-00580-6

Keywords

Navigation